MR. J. J. WALKKR ON THE DIAMETERS OF A PLANE CUBIC. 159 



or (15) 



4 A'*{D- - (Ds) 3 } = 4{u D 2 u - (Du 2 )}6 D 2 { D 2 u - (Du) 2 } 



-[4D{uD 2 -(Dw) 2 }] s (18) 



But 



4 D{tt D 2 w - (D) 2 } = 3 Du D 2 u + u D 3 u - 4 Da Du 



= uD* DuD*u; (19) 



and 



12 D 2 {u D 2 n - (Du) 2 } = 3 DM D 3 u - 2 (D 2 w) 2 - D D 3 ?*, 



6 D 8 { D'-w - (D) 2 } = D D3u - (D 2 u) 2 (20) 



By substitution from (19) (20) in (18) 



4 A'<{ D 2 * - (D*) 2 } = 4{u D 2 w - (D) 2 } {Du D 3 u - (D 2 w) 2 } 



- (u D 3 * - Du D 3 u) 2 , 

 whence (16) (17) 



...), ... (21) 



i.e., the condition that L shall touch s, is equal to one-fourth of that for L touching the 

 culnc u, with changed sign . (i) 



22. Considering next the discriminant of s; viz. (9), 



if, for shortness (7), 



v l = Dt*i, v 3 = Dug, ig = Dw 3 , T * 



W L = D 2 !^, u> 2 = D 2 Wj, w 3 = D 2 t/ 3 , J 



V, = -U7 3 3 tt' 3 W;j, V 2 = W S U 1 W\U 3 , V 3 =: M>jt* 2 W' 2 1( ^ . (24) 



then (13, 14), 



A'*(4t* is w 33 Ujg 3 ) = 4(u 3 u> i - r a 8 ) (w 3 u> 3 - r 3 2 ) - ( 2 w s -f- u.,u> 3 2ty> 8 ) 3 



= 4U 1 W 1 -V 1 2 ; . . (25) 



* By means of these expressions the equation of the poloid () may be thrown into a form which 

 exhibits it explicitly as the envelope of the polar line of points in L: viz., by (9), (13), (14), (23), 



Of the right lines in this form, (i) xu\ + ... is the polar line of the point x'y'z' in L; (ii) xw l -f ... or 

 (.11 f) *i + is the Newtonian diameter of chords of u parallel to L (shown in fig. 1 touching the 

 poloid at D'); (iii) jn\ + ... or x'v l + ... is the chord of contact with () of (i), (U). (April, 1888.) 



